FIRST-PRINCIPLES CALCULATIONS OF THE ELECTRONIC STRUCTURE OF GRAIN BOUNDARIES

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FIRST-PRINCIPLES CALCULATIONS OF THE ELECTRONIC STRUCTURE OF GRAIN

BOUNDARIES

E. Sowa, A. Gonis, X.-G. Zhang

To cite this version:

E. Sowa, A. Gonis, X.-G. Zhang. FIRST-PRINCIPLES CALCULATIONS OF THE ELECTRONIC

STRUCTURE OF GRAIN BOUNDARIES. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-

335-C1-340. �10.1051/jphyscol:1990153�. �jpa-00230314�

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Colloque Cl, suppl6ment au n o l , Tome 51, j a n v i e r 1990

FIRST-PRINCIPLES CALCULATIONS OF T H E ELECTRONIC STRUCTURE OF GRAIN BOUNDARIES

E . C . SOWA, A. GONIS and X.-G. ZHANG"

Lawrence Livermore National Laboratory, L356 Livermore, CA 94550, Y . S . A .

Physics Department, Northwestern University, Evanston, IL 60201, U.S.A.

Abstract

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We have developed a new, first-principles method for the calculation of the electronic structure of surfaces, grain boundaries, and other low-symmetry systems. Our method yields the electronic density-of-states (DOS) for unrelaxed as well a s structurally-relaxed grain boundaries. I t is based on a real-space

formulation of multiple-scattering theory (RSMST) and thus does not rely on perfect periodicity and lattice Fourier transforms. This method allows us to bridge the gap between atomistic calculations of relaxed grain-boundary configurations, as obtained within the Embedded Atom Method (EAM), and first-principles quantum-mechanical calculations. We present results for realistic twist and tilt grain boundaries in metals, and discuss our plans for future development.

Electronic-structure calculations require two ingredients: Knowledge of the atomistic structure of a material, and the existence of methods, both formal and computational, for treating the structure a t hand. I n the case of materials with full translational symmetry, e.g. bulk'crystalline solids, both requirements can be satisfied.

The atomistic structure may be determined by, e.g., X-ray crystallography. Once this is known, the electronic structure may be obtained with the aid of Bloch's theorem and the associated lattice Fourier transforms, which diagonalize the Hamiltonian i n reciprocal space (k-space). In contrast, the determination of both the atomistic and electronic structure of materials i n which translational symmetry is broken by surfaces or internal interfaces is much more problematic. Experimental determination of the atomistic structure of some surfaces is possible through, e.g., low-energy electron diffraction in reciprocal space or scanning tunneling microscopy in real space, but there are no techniques available for imaging atomic positions a t internal interfaces. In some cases, partial information may be obtained with electron microscopy, but often one must rely on

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990153

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Because nearly all existing, fully first-principles formalisms for performing electronic-structure calculations are based on the properties of translational invariance and the use of Bloch's theorem, knowledge of atomic coordinates 0f.a surface or interface does not necessarily enable such calculations to be performed with conventional methods.

Consequently, it has often been found necessary to invoke rather severe approximations with respect to the structure of a system, such as the use of slabs of finite thickness to treat surfaces, or of repeating slabs or supercells to study internal interfaces. Such approximations are highly undesirable for a number of reasons: First, they are

"uncontrolled", yielding results which should be checked for dependence on slab thickness on a case-by-case basis. Second, they are conceptually unattractive, involving rather severe approximations to the underlying geometry of the system. Third, the extent to which they may yield accurate results may lull one into a false sense of

accomplishment, obscuring the existence of a still unsolved problem.

Stated briefly, this problem consists in finding the solution of the one-particle Schrodinger equation, within the Born-Oppenheimer approximation (fixed nuclei) and the local-density approximation to density-functional theory, that satisfies the proper boundary conditions imposed by the structure of a particular semi-infinite material.

In a recent publication 151, a first-principles, multiple-scattering formalism, which allows the exact treatment of the problem just stated, was introduced. As the formal aspects of this real-space multiple-scattering theory (RSMST) method have been reviewed in previous work 15, 61, we shall forgo all but a brief description of this method here. The RSMST is based on the concept of semi-infinite periodicity (SIP), defined as the regular repetition along a given direction of a scattering unit (atom, planes of atoms, etc.), or a set of such units. Systems with SIP possess the property of removal invariance, which states that the scattering properties (scattering matrices) of any such system remain invariant when an integral number of scattering units is removed from, or added to, the free end of the system. Using this property in conjunction with multiple-scattering theory, one can determine the electronic Green function, and hence all one-particle quantities such as the density-of-states (DOS), directly in real space, bypassing often cumbersome reciprocal- space (k-space) integrations. This formalism provides a unified treatment of the electronic properties of a broad spectrum of systems that includes, but is not limited to, pure elemental solids, compounds, ordered alloys, surfaces and interfaces, and other low-symmetry systems. Only systems with no recognizable periodic structure, e.g.

amorphous materials and liquids, fall outside the scope of the RSMST method.

The essence of the method consists in a prescription for the proper renormalization of the scattering properties of the boundary sites of a cluster of atoms. Unrenormalized or

"bare" sites in the interior of the cluster describe the region of interest, such as a grain boundary, while the renormalized boundary sites represent the infinite medium

surrounding the cluster. This "dressing up" of the cluster is done independently for each part of a system that is characterized by its own SIP, so that grain boundaries between essentially arbitrary crystal structures can be treated. At its present stage of development, our codes can be applied to known atomistic configurations with known electronic one- particle potentials. Work currently in progress is aimed a t the incorporation of the often important effects of charge self-consistency, and of total-energy capability, into the program.

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^Method

In the calculations reported here, we used the self-consistent potentials for bulk Cu given by Moruzzi, Janak and Williams (MJW) 171. The atomic coordinates of the

unrelaxed grain boundaries are easily found through an appropriate twisting or tilting of one half of the underlying lattice. We used unpublished EAM calculations, performed by S. M. Foiles of Sandia National Laboratories, Livermore, to obtain the atomic coordinates of relaxed twist and tilt grain boundaries in Cu.

I11

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R e s u l t s

Figure 1 displays the local electronic DOS'S corresponding to a coincidence site (a site which is common to the lattices on both sides of an interface) at both a n unrelaxed and a relaxed C5 (100) 36.9' twist grain boundary in Cu. The bulk Cu DOS, calculated

with the RSMST, is also displayed for comparison. (This bulk DOS reproduces the main features of the MJW calculation; small differences may be attributed to our use of a coarser energy grid and a small imaginary component of the energy.) The unrelaxed grain-boundary DOS exhibits considerable smearing of structure compared to the bulk DOS, because of the loss of periodicity and the associated destruction of the Van Hove singdarities. The grziin boundary DOS is slightly broader than that of the bulk material due to the decreased distance between some of the Cu atoms across the interface. (In fact, simply twisting one half of the crystal with respect to the other results in some of the atoms overlapping across the boundary).

iii

_V

C

Fig. 1.

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Electronic density-of-states associated with a coincidence site a t a Z5 (100) 36.9' twist grain boundary in Cu. DOS'S for the unrelaxed and relaxed configurations are shown, respectively, as solid curves in panels (a) and (b), and are compared with those of

bulk Cu (dotted curves).

Bath the atomistic and electronic characteristics change noticeably in the case of the relaxed grain boundary. The results of the EAM calculations indicate that the

interplanar spacing increases by about 20% across the interface from its bulk value, decreases by 2% in the next set of layers, and remains essentially unchanged i n layers deeper inside the material. The increase is the result of relieving the overlap conditions across the interface mentioned above, and its effects on the electronic structure are similar to those associated with decreased coordination. In these first calculations, we included only the 20% expansion a t the boundary layer, which is the dominant effect. It is seen that the DOS a t the ^grain boundary is indeed narrower than that of bulk Cu. It is also seen that the relaxed grain boundary DOS is shifted slightly toward lower energies compared both to that of bulk Cu and of the unrelaxed configuration, and that it possesses

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somewhat sharper structure than the DOS at an unrelaxed grain boundary. Although the present, non-charge-self-consistent calculation cannot provide reliable information about the relative energies of the various configurations, both of these effects are consistent with the lower energy of the EAM-relaxed configuration with respect to the unrelaxed one.

a

boundary plane

Fig. 2.

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Side view of the atomistic arrangements of a C5 (310) tilt grain boundary in Cu, with filled and unfilled circles representing atoms on two different (100) planes.

Unrelaxed configuration, panel (a); relaxed configuration, panel (b).

In order to make clearer the discussion of tilt grain boundaries, it is helpful to refer to the schematic diagram of Fig. 2, which shows the atomic arrangements at a C5 (310) tilt grain boundary in Cu. The letter A designates a coincidence site, which remains relatively immobile through the relaxation process, while B denotes a site that according to the EAM calculations is displaced the most during relaxation. The relaxations of layer spacings predicted by the EAM for six layers on either side of the boundary were included in this calculation; the primary relaxation is the movement of atom B to a position coplanar with atom A, forming a mirror plane at the boundary. The local DOS'S a t site A in both the unrelaxed and the relaxed configurations are shown in Fig. 3, and are compared with those of bulk Cu. It is interesting to note that the bulk DOS peak a t 4.5 Ry persists in the (100) twist boundary but is absent from the (310) tilt boundary. From this we may conjecture that this structure results primarily from the two-dimensional periodicity of (100) layers. Analogous results for site B are shown in Fig. 4. The changes in the tilt-

boundary DOS's upon relaxation are qualitatively similar to those of the twist grain boundary discussed above. (The rather high DOS'S a t the low-energy region

corresponding to site B is most probably an artifact of the scheme used to integrate the cell wave functions, required /5,6/ for calculating the DOS, over the cell polyhedron whose shape in this case deviates substantially from that of the undistorted lattice). We note that these DOS's resemble in structure those obtained in other work /8/ where a C5 (210) tilt grain boundary was studied.

Fig. 3.

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Results analogous to those depicted in Fig. 1, but for a coincidence site in a Cu

E 5

(310) tilt grain boundary (site A in Fig. 2).

V

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Conclusion

As the calculations just presented illustrate, the RSMST method allows one to obtain the electronic structure of materials with extended defects, such as surfaces and

interfaces, on an atom-by-atom basis. Although in its present stage our code is slower than those based on conventional methods when applied to systems of high symmetry, it holds the distinct advantage of being applicable to low-symmetry structures that are not amenable4to treatment by these methods. We are currently attempting to increase the efficiency of our code and add charge self-consistency and total-energy capabilities. The results of these efforts will be communicated as they become available.

Aclmowledgment

This work was performed under the auspices of the Division of Materials Science of the Office of Basic Energy Sciences, U. S. Department of Energy, and the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48.

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Fig. 4.

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Results analogous to those shown i n Fig. 3 but for a displaced site (site B in Fig. 2).

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a

(1984) 6443.

Foiles, S. M., Baskes, M. I. and Daw, M. S., Phys. Rev. B

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^{(1986) 7983.}

Foiles, S.

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Acta Metall. (1989) , to be published.

Foiles, S. M., i n Characterization of the Structure and Chemistry of Defects in Materials, eds. Larson, B. C., Riihle, M. and Seidman, D. N., (Materials Research Society, Pittsburgh, 1989).

Zhang, X . 4 . and Gonis, A., Phys. Rev. Lett. Q (1989) 1161.

Zhang,

X.-G.,

Gonis, A. and Maclaren, J. M-, Phys. Rev. B. (1989)

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^to be published.

Moruzzi, V. L., Janak, J. F. and Williams, A. R., Calculated Electronic Properties of Metals, (Pergamon Press, New York, W , 1978).

Crampin, S., Vvedensky, D. D., Maclaren, J. M. and Eberhart, M. E., Phys. Rev. B.

(1989)

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to be published.